Maths Module 4 - James Cook University

Maths Module 4

Powers, Roots and Logarithms

This module covers concepts such as: ? powers and index laws ? scientific notation ? roots ? logarithms

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Module 4

Powers, Roots, and Logarithms

1. Introduction to Powers 2. Scientific Notation 3. Significant Figures 4. Power Operations 5. Roots 6. Root Operations 7. Simplifying Fractions with Surds 8. Fraction Powers/Exponents/Indices 9. Logarithms 10. Helpful Websites 11. Answers

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1. Introduction to Powers

Powers are a method of simplifying expressions. ? An equation such as : 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 = 63 could be simplified as: 7 ? 9 = 63 ? Whereas an expression such as: 7 ? 7 ? 7 ? 7 ? 7 ? 7 ? 7 ? 7 ? 7 could be simplified as: 79

A simple way to describe powers is to think of them as how many times the base number is multiplied by itself.

? 5 ? 5 ? 5 ? 5 is the same as 54 is the same as 625

54

? The 5 is called the base. ? The 4 is called the exponent, the index or

the power.

? The most common way to describe this expression is, `five to the power of four.'

? The two most common powers (2 & 3) are given names: 32 is referred to as `3 squared' and 33 as `3

cubed.'

? However, note that -32 is different to (-3)2; the first is

equivalent to (-9), whereas the second is equivalent to (+9) In the first example, -32, only the 3 is raised to the power of

two, in the second, (-3)is rasied to the power of two, so

(-3) ? (-3) = 9, `positive' number because (-) ? (-) = (+)

? Whereas, a negative power, such as 6-3, reads as `six raised to

the power of negative three', it is the same as 613: it is the

reciprocal.

6-3

=

1 63

=

1 216

? To raise to the negative power means one divided by the base raised to the power, For example, 6-3

can be read as `one divided by the 6 raised to the power of three' (six cubed).

?

Another example:

1 43

is

the

same

as

4-3

,

which

is

also

1 64

? Take care when working with powers of negative expressions: (-2)3 =

(-8) = (- ? - ? -= -) (-2)4 = 16 (- ? -)(- ? -) = + (-)5 will be a negative expression, whereas (-)6 will be positive.

It helps if you can recognise some powers, especially later when working with logarithms The powers of 2 are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 2048, 4096, 8192,... The powers of 3 are: 3, 9, 27, 81, 243, 729,... The powers of 4 are every second power of 2 The powers of 5 are: 5, 25, 125, 625, 3125,... The powers of 6 are: 6, 36, 216, 1296,... The powers of 7 are: 7, 49, 343, 2401,... The powers of 8 are: every third power of 2 The powers of 9 are every second power of 3 The powers of 10 are: 10, 100, 1000, 10 000, 100 000, 1 000 000,... The powers of 16 are every fourth power of 2

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1. Your Turn:

Work out the value of the following based on what you have understood from the introduction:

a. 23 =

i. (0.5)3 =

b. 92 =

j. 4-2 =

c. 123 =

d. (-4)2 =

e. (-3)3

f.

(-

1)2

4

=

g. 03 =

h. (-0.1)2 =

k. 1-11 =

l. 4-1 =

m. (-4)-1 =

n. (0.5)-4=

o.

(3)-3 =

4

p. (-2)-3 =

4

2. Scientific notation

? The most common base is 10. It allows very large or small numbers to be abbreviated. This is

commonly referred to as `scientific notation'. o 105 = 10 ? 10 ? 10 ? 10 ? 10 = 100 000. (Note: 100 000 has 5 places (as in place

value) behind `1', a way to recognise 10 to the power of five

o

10-4=

1 104

=

10

?

10

?

10

?

10

=

0.0001. (Note: 0.0001 has 4 places (as in place value) in

front; one divided by 10 to the power of four)

o 3000 = 3 ? 1000 = 3 ? 103

o 704 500 000 = 7.045 ? 100 000 000 = 7.045 ? 108

? A classic example of the use of scientific notation is evident in the field of chemistry. The number of molecules in 18 grams of water is 602 000 000 000 000 000 000 000 which is written as 6.02 x 1023 which is much easier to read and verbalise.

2. Your Turn:

Write the following in scientific notation: a. 450

b. 90000000

c. 3.5

d. 0.0975

Write the following numbers out in full: e. 3.75 ? 102 f. 3.97 ? 101 g. 1.875 ? 10-1 h. -8.75 ? 10-3

3. Significant Figures

Scientific notation is used for scientific measurements as is significant figures. We briefly touched on this concept in module one. Both scientific notation and significant figures are used to indicate the accuracy of measurement. For example, if we measured the length of an item with an old wooden ruler, a steel ruler, and then some steel callipers, the measurements would vary slightly in the degree of accuracy.

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